Given a family $\mathcal{B}$ of boxes parallel to the axis of $\mathbb{R}^d$, let $\tau$ be its penetration number and $\nu$ its independence number. Whether $\tau/\nu$ can be arbitrarily large given $d\geq 2$ is an old question. where $$\tau\geq \Omega_d(\nu)\cdot\left(\frac{\log \nu}{\log\log \ nu}\right)^{d -2}.$$ This not only answers the previous question affirmatively for all $d\geq 3$ , but it also matches the best known upper bound up to the double-log factor.
Our main structure also suggests further about the Ramsey and coloring properties of the composition of the boxes. We show that there exists a family of $n$ boxes in $\mathbb{R}^{d}$ whose intersection graph is the clique and the independent number $O_d(n^{1/2})\cdot \left(\ frac{\log n}{\log\log n}\right)^{-(d-2)/2}.$ This is the first improvement over the trivial upper bound $O_d(n^{1/2}) is. $, and match the best known lower bounds up to the double logarithmic coefficient. Finally, for each $\omega$ satisfying $\frac{\log n}{\log\log n}\ll \omega\ll n^{1-\varepsilon}$, the intersection of $n$ boxes Create a chart. clique number is at most $\omega$, chromatic number $\Omega_{d,\varepsilon}(\omega)\cdot \left(\frac{\log n}{\log\log n}\right)^{ d- 2}.$ This matches the known upper bound of $O_d((\log w)(\log \log n)^{d-2})$ times.
